On the matrix algebra realization of the theory of biquaternions

This paper describes a matrix algebra presentation of Clifford's theory of biquaternions. We examine 4 × 4 skew-symmetric matrices and use their exponentials to relate quaternions to equal-angle double rotations in Euclidean four-space E⁴. We show how double rotations in E⁴ expressed in terms of plane coordinates lead to elliptic biquaternions in both Plucker and Study forms and present the fundamental Plucker and Study conditions that govern the biquaternions. Finally, we show that a spatial displacement in E³ in terms of parabolic biquaternions (or dual quaternions) is a limiting case of a double rotation in E⁴ in terms of elliptic biquaternions.